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Russia Izhevsk
Section Mathematics
Title The discrete Schrödinger equation for a quantum waveguide
Author(-s) Tinyukova T.S.a, Chuburin Yu.P.b
Affiliations Udmurt State Universitya, Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesb
Abstract We investigate the spectral properties of the discrete Schrödinger operator for the infinite band with zero boundary conditions. We prove that the eigenvalues and resonances arise for the small decreasing potentials near singularities of the non-perturbed Green function (boundary points of the subbands) and we find their asymptotic behavior. The scattering picture is described: the diffraction (i.e. the scattering mainly in the finite number of preferential directions) transforms into probability waves in time of the reflection and propagation in the considered quasi-1D system. The simple formulas for these probabilities are obtained near boundary points of the subbands (this corresponds to small velocities of the quantum particles) for the small potentials.
Keywords discrete Schrödinger operator, quantum waveguide, eigenvalue, resonance, transmission and reflection coefficients
UDC 517.958, 530.145.6
MSC 81Q10, 81Q15
DOI 10.20537/vm120407
Received 10 September 2012
Language Russian
Citation Tinyukova T.S., Chuburin Yu.P. The discrete Schrödinger equation for a quantum waveguide, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, issue 4, pp. 80-93.
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